On the non-existence of oscillation numbers in Sturm-Liouville theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909185595146240 |
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| author | Mingarelli, Angelo B. |
| author_facet | Mingarelli, Angelo B. |
| contents | We prove an old conjecture that relates the existence of non-real eigenvalues of Sturm-Liouville Dirichlet problems on a finite interval to the non-existence of oscillation numbers of its real eigenfunctions, [[6], p.104, Problems 3 and 5]. This extends to the general case, a previous result in [1], [2] where it was shown that the presence of even one pair of non-real eigenvalues implies the non-existence of a positive eigenfunction (or ground state). We also provide estimates on the Haupt and Richardson indices and Haupt and Richardson numbers thereby complementing the original Sturm oscillation theorem with the Haupt-Richardson oscillation theorem discovered over 100 years ago with estimates on the missing oscillation numbers of the real eigenfunctions observed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_19575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the non-existence of oscillation numbers in Sturm-Liouville theory Mingarelli, Angelo B. Classical Analysis and ODEs Mathematical Physics Quantum Physics 34B24, 34L10, 34C10 We prove an old conjecture that relates the existence of non-real eigenvalues of Sturm-Liouville Dirichlet problems on a finite interval to the non-existence of oscillation numbers of its real eigenfunctions, [[6], p.104, Problems 3 and 5]. This extends to the general case, a previous result in [1], [2] where it was shown that the presence of even one pair of non-real eigenvalues implies the non-existence of a positive eigenfunction (or ground state). We also provide estimates on the Haupt and Richardson indices and Haupt and Richardson numbers thereby complementing the original Sturm oscillation theorem with the Haupt-Richardson oscillation theorem discovered over 100 years ago with estimates on the missing oscillation numbers of the real eigenfunctions observed. |
| title | On the non-existence of oscillation numbers in Sturm-Liouville theory |
| topic | Classical Analysis and ODEs Mathematical Physics Quantum Physics 34B24, 34L10, 34C10 |
| url | https://arxiv.org/abs/2404.19575 |